The Hoops went into the second-leg of their qualifying tie trailing 2-0 but Kris Commons and Georgios Samaras levelled the aggregate with goals either side of half-time before James Forrest scored the winner three minutes into stoppage time.
Karagandy spent much of the build-up to the Wednesday's clash dismissing Celtic's chances, and Brown admits their approach fired up the Scottish Champions.
Brown said: "That was sweet. They came here saying they expected to score two or three but everybody underestimated us and we showed how good a team we really are. We were angry about everything last week.
"The only shot they had on goal tonight came from a long throw-in when they hit the bar. That's all they have: long throws.
"I got booked for celebrating the way they celebrated at the end of the first game. They were in everybody's faces.
"I know we are a big club and they were delighted to beat us but you have to show a bit of respect and they got what they deserved."
Celtic striker Anthony Stokes, who was involved in both second half goals, echoes Brown's sentiments.
Stokes said: "Listen, they did their talking in the papers; we did our talking on the pitch tonight. It's as simple as that.
"We rode our luck. One came off the crossbar. But that was always going to be the case; they had a game plan and it wasn't going to change. They played long balls into the box, hoofed balls up the pitch."